Prime Numbers

80 Chapter 1 PRIMES!
(see [Bach 1997b]). Then, noting that a certain part of the infinite sum is
essentially the Euler constant, in the sense that
∞
n ζ(n) − 1
γ +ln2−1= (−1) ,
n
n=2
use known methods for rapidly approximating ζ(n) (see [Borwein et al. 2000])
to obtain from this geometrically convergent series a numerical value such as
B ≈ 0.26149721284764278375542683860869585905156664826120 ....
Estimate how many actual primes would be required to attain the implied
accuracy for B if you were to use only the defining product formula for B
directly. Incidentally, there are other constants that also admit of rapidly
convergent expansions devoid of explicit reference to prime numbers. One of
these “easy” constants is the twin prime constant C2, as in estimate (1.6).
Another such is the Artin constant
A =
1 −
p
1
p(p − 1)
≈ 0.3739558136 ...,
which is the conjectured, relative density of those primes admitting of 2 as
primitive root (with more general conjectures found in [Bach and Shallit
1996]). Try to resolve C2, A, or some other interesting constant such as
the singular series value in relation (1.12) to some interesting precision but
without recourse to explicit values of primes, just as we have done above for the
Mertens constant. One notable exception to all of this, however, is the Brun
constant, for which no polynomial-time evaluation algorithm is yet known. See
[Borwein et al. 2000] for a comprehensive treatment of such applications of
Riemann-zeta evaluations. See also [Lindqvist and Peetre 1997] for interesting
ways to accelerate the Mertens series.
1.91. There is a theorem of Landau (and independently, of Ramanujan)
giving the asymptotic density of numbers n that can be represented a 2 + b 2 ,
namely,
#{1 ≤ n ≤ x : r2(n) > 0} ∼L x
√ ln x ,
where the Landau–Ramanujan constant is
L = 1
√
2
1 − 1
p2 −1/2 =0.764223653 ...
p≡3 (mod4)
One question from a computational perspective is: How does one develop
a fast algorithm for high-resolution computation of L, along the lines, say,
of Exercise 1.90? Relevant references are [Shanks and Schmid 1966] and
[Flajolet and Vardi 1996]. An interesting connection between L and the
possible transcendency of the irrational real number z =
n≥0 1/2n2 is found
in [Bailey et al. 2003].
1.92. By performing appropriate computations, prove the claim that the
convexity Conjecture 1.2.3 is incompatible with the prime k-tuples Conjecture